isolated non-normal surface singularity

Identify two points in the affine plane over $\mathbb C$: it is the affine variety given by the algebra $$\{ f\in\mathbb C[x,y] \mid f(0,0)=f(0,1)\}.$$ Its normalization is the affine plane.

If $k$ is a field and $A=k[x,xy,y^2,y^3]\subset k[x,y]$, then $X=Spec (A)$ is a surface with the maximal ideal $\langle x,xy,y^2,y^3\rangle \subset A$ as only (and thus isolated) singularity.
Thus $X$ is non normal because the domain $A$ is not integrally closed.
Here are some details:

I) $A$ is not integrally closed because $y\in Frac(A)\setminus A$ is integral over $A$ (it satisfies $T^2-y^2=0$)
II) The morphism $$f:\mathbb A^2_k \to \mathbb A^4_k:(x,y)\mapsto (u=x,v=xy,w=y^2,z=y^3) $$ has as image $X$, the surface $X\subset \mathbb A^4_k$ defined by the three equations $$ u^2w=v^2,\: u^3z=v^3, \:w^3=z^2 $$ III) If we put $O=(0,0)\in \mathbb A^2_k$ and $O'=(0,0,0,0) \in X\subset \mathbb A^4_k$, the morphism $f$ restricts to an isomorphism $f_0:\mathbb A^2_k\setminus \lbrace O \rbrace \stackrel {\cong}{\to} X\setminus \lbrace O'\rbrace$.
Its inverse $$f_0^{-1}:X\setminus \lbrace O'\rbrace \stackrel {\cong}{\to} \mathbb A^2_k\setminus \lbrace O \rbrace: (u,v,w,z)\mapsto (x,y)$$ is given by: $$x=u \\ y=v/u \;\text {if } u\neq 0 \quad \text {or} \quad y=z/w \; \text {if } w\neq 0$$
Edit
The surface $X$ is not isomorphic to QiL's beautifully simple example since the normalization map for $X$ is bijective, and it isn't for QiL's surface.
However Justin remarks that my $A$ can be described as the ring of polynomials $f(x,y)\in k[x,y]$ such that $f_y(0,0)=0$, whereas QiL requires $f(0,0)=f(0,1)$
I find this analogy very interesting and I am grateful to Justin for having pointed it out, since I had not noticed it at all.